Before I share my favourite links from the last month, I have a little personal news to share. I spent the period from the 24th to the 27th of September catching up with family, including relatives from other parts of the country. Each day had its own special event — a school concert and school play that some of my relatives were part of, a seven-year-old’s birthday party, and my nephew’s baptism.

Here is an ultra-cute video of my niece (in the pink) with a friend on the trampoline. It was recorded at my sister’s place at lunch on the 27th.

I liked this idea: Venn diagram poetry, mostly because I think it would be an interesting challenge to write your own.

Thirty minute radio segment on selective mutism. I have more to say about this below.

Enlightening article on the cultural context of the British piggate scandal.

I’d like to add a few comments to the item on selective mutism. It’s not something I’ve experienced directly, but learning about other people’s experiences — through documentaries and so forth — always takes my thoughts and emotions in interesting directions. It evokes memories of experiences that, while not the same as selective mutism, serve as analogies that I can draw on to understand it better. And it makes me fantasise about what I’d say to the people whose stories I hear if I could meet them in their past.

I’ll share one memory as an example. I was raised in a religious household, and during my teenage years I was Christian myself, but I never joined in the ritual of saying grace before a meal. I remember one day when my parents expressed their wish that I would, to which I said something like “I think I could if– if– if– …” and faltered. My parents reacted poorly to that, telling me I shouldn’t bargain with them, but the words I couldn’t get out that day were: “I think I could if you promise not to overreact, not to make a big deal out of it, not to make me feel like the centre of unwanted attention.”

When I think of selective mutism, I think of that memory and others like it, and multiply them by a thousand in my mind. The analogy is far from perfect, but it’s something — a seed of connection around which further empathy and understanding can be built.

I also think of the song Across the Waters by Jimmy Gregory (from the 1996 album West Along the Road). The song is really about lovers who are separated geographically, and celebrates the fact that, however much they miss each other, their love is strong enough to withstand being apart. But I feel the following excerpt could just as easily be about selective mutism, and in that context is extremely poignant:

There’s a strength in the silence between usStill waters run deep.There’s an ocean of words that I’d say to youBut sure all of them will keep.

In those words I hear an acknowledgement of the turmultous emotions and intense desire to communicate that lies beneath the silence of selective mutism, along with an assurance that there is no pressure: that it is OK if today is not a day when words can be spoken. Do you agree? What do they evoke for you?

If I was trying to build rapport with someone suffering from selective mutism — trying to create an environment where they could feel comfortable and understood — then between the song lyrics, the memories, and the willingness to learn, I like to think I’d have something to offer. Though the opportunity to show it would come less easily in life than in my fantasies. Comments will be gratefully received, especially from readers who have been there.

Over the years I’ve been blogging, I’ve found that leafing through the contents of my desk can be a rich source of inspiration. My desk drawers contain records of my own creativity from the past, as well as other sorts of memories, and I’ve written many blog posts based on that material.

But there is a tendency for miscellaneity to fall through the cracks. If something is particularly significant there’s a good chance I’ve already blogged about it, whereas other items have always seemed too trivial to share even though they’ve been in storage for years.

Today I’m going to share three of them. They may be trivial, but there’s no telling what will resonate with the interests of readers and inspire an interesting conversation in the comments. That would make them well worth including on the blog.

1. A querying language invented while at university.

At university I majored in computing, which naturally included a subject or two about databases — SQL, relational algebra, and all that stuff. Elsewhere in the course I learned to read and write in EBNF notation.

For my own amusement I sometimes toyed with inventing my own programming languages — not implementing them (inventing and implementing are two very different things), but figuring out how the syntax would work and writing documentation for them as though they actually existed. In a sense it was my way of consolidating what I was learning, especially about trade-offs between different programming languages and things like that.

One of my inventions was a querying language with the same functionality as SQL, but a syntax based more consistently on relational algebra. I’ve uploaded a copy of my documentation. It may be complete rubbish, but over a decade later it hardly matters.

2. An email exchange about sensory integration.

This is another item from my university days, but not connected to my studies.

At an autism conference many years ago, I bought a textbook on sensory integration, which interested me enough to write an email to one of the authors. I described some of the games I played as a child, and speculated that (to paraphrase for the blog) there may be a correlation between people who are prone to motion sickness and people who find it hard to keep a tidy room.

My hypothesis was that since picking items off the floor involves continual head altitude changes, people who struggle with it might often be those who find it hard to modulatevestibular sensations while simultaneously concentrating on a goal-directed task.

The textbook author replied, saying: “I was also interested in your comments about tidying up and vestibular/modulation difficulties. I have certainly noticed that tendency in children but none has ever explained it quite so graphically. I certainly will think about that.”

Or, as I once paraphrased it in a comment thread, she said that the idea was plausible and consistent with anecdotal observations.

3. A shortlist of aesthetically selected Arabic male names.

I’ve browsed the baby names website behindthename.com on numerous occasions. One time, on a whim, I decided to browse the section on male Arabic names, just to see what I liked the sound of. I was particularly looking for pairs of names that sound good together, consciously ignoring the fact that Western naming practices (first name plus middle name) are not usual in Arabic culture.

Filed away in my desk drawer and certain locations on my computer, I still have various items from the days when Christian faith was a part of my identity. These include:

Some evangelistic web pages I wrote;

Religious discussions from online forums;

A notebook for jotting down theological speculations;

Various other items.

It’s been a few years since I’ve mentioned my religious past on this blog, but see my 2009 post about the afterlife and my 2011 post about prayer. As I aim to blog on a variety of subjects, now seems a good time to dig into those archives in search of an idea for a new post.

Since Easter wasn’t long ago, why not look at what my thoughts were, when I was a Christian, on the death and resurrection of Christ?

There is more diversity than often admitted, within thoughtful Christian communities, about what the crucifixion of Christ has to do with the reconciliation of sinners and God. Christians agree that the crucifixion was necessary, but there are different explanations for why (the New Testament’s account is largely metaphorical). Unfortunately, certain interpretations are more culturally dominant than others and the fact the discussion exists tends to be obscured. And of course, many people are content to accept the necessity on faith and leave the theologising to others.

The most familiar explanation is that it is an aspect of God’s nature that he literally cannot allow sin to go without its punishment, and that in the death of Christ he reconciles that necessity with the demands of his love by taking the punishment upon himself. We have all been exposed to that idea — not least from the allegories of C. S. Lewis — and there are many who insist on it as a defining article of Christian faith. But in all my years as a Christian, that viewpoint never made any sense to me, and it was never a part of my theology.

A less widespread explanation — but one with its share of advocates — is that the crucifixion was at heart the ultimate demonstration of God’s love. According to this idea, the normal expectation of humanity is to see God as judging and demanding, to be preoccupied with earning God’s acceptance and fearful of his rejection. And so Jesus comes to offer humanity a different view of God: a God who will forgive us even if we crucify him, a God whose love is (quoting Geoff Bullock) “beyond humankind’s capacity to earn it”.

That explanation also never felt like it could be the whole answer — largely because that view of God is clearly not as universal as it makes out — although I could envisage it as part of the answer. Anyway, those were the two competing explanations I found in books,* and, finding them inadequate, I speculated.

One line of thought I followed was that before a person can be healed of their sinful condition and fitted for Heaven, it is necessary to first develop a visceral appreciation of how serious sin is. That is to say, God can no more operate on the soul of someone who is insufficiently horrified by immorality, than a dentist can fix the tooth of someone who won’t open their mouth wide enough. And by bearing in mind the image of Jesus on the cross — the most perfect life ever lived tortured to death for the approval of the crowd — and learning to associate that image with all sin including one’s own seemingly trivial transgressions, it’s possible to put sin into its proper perspective and prepare the human heart to be transformed by God.

The main problem I acknowledged was that this doesn’t explain why some other illustration of the enormity of sin shouldn’t do just as well: why should the crucifixion and only the crucifixion be adequate to elicit the proper level of repulsion? I had no answer to that, but it seemed plausibly on the right track.

A further speculation which unites some of the above ideas is best presented as a metaphor, like the Scylla and Charybdis of Greek mythology. In this account there are two opposing obstacles on the spiritual journey, both of which must be negotiated if one is to make it to Heaven. The first is the danger of thinking our moral imperfections are no big deal — that we are good enough without God’s intervention — which in Christian thought is often looked upon as the ultimate folly. The second is the danger of being so overcome by a sense of our own inadequacy that we imagine God would never accept us, and are afraid to make the approach. Above this landscape — a narrow passage between two opposing and terminal errors — the Cross shines like a lighthouse, warning us at one and the same time not to underestimate sin, but also not to underestimate forgiveness.

That is as good an answer as I ever had. It works pretty well as a metaphor, but (just like every other explanation) not so well under clinical observation. In typical mythological style, the obvious questions — like why one universal lighthouse is better than several smaller ones — can be answered only by appealing to the seductive simplicity of the story. I realised that my answers were inadequate, even in conjunction, but they reinforced the impression that there was an answer to be found. At the end of the day, though, I accepted the idea that the crucifixion was somehow necessary for humans to be fitted for Heaven, even if I didn’t see why.

No religious belief can be understood in isolation, and everything I’ve said relates to topics that are outside the scope of this post (for example, this is not the time to discuss how I understood the identity between Jesus and God). But I think it’s worth saying that I always believed God shares all human suffering — like a divine mirror-touch synaesthete — and that I was absolutely sure no-one’s destiny hinges on ideas they may or may not encounter in this lifetime (as stated in my 2009 post).

We are all shaped by beliefs we no longer hold, and no doubt I wouldn’t be the same person today if I hadn’t in the past been a devoted believer. I am convinced of the value of occasionally discussing our former convictions not to defend nor refute them, but in the same spirit that one might share an old photograph. I hope this reminiscence has contained some thought for everyone to take away, Christians and atheists and all others as well.

* For further reading, see The Plain Man Looks at the Apostle’s Creed by William Barclay. (Also, Power of Your Love — Jesus: The Unexpected God by Geoff Bullock, although that book is irritatingly prone to speculation masquerading as fact.)

I vaguely remember wondering — as a child of seven or eight — how new mathematics became established. I pictured a long line of people in front of a guy with a desk (or maybe a tent), waiting to register their latest idea — such as a new mathematical operation.

I had one such idea myself. I knew by this age that multiplication consisted of repeated addition, so I thought why not define something else in terms of repeated multiplication? Thus I invented integer exponentiation. (It’s an obvious idea: I blame the Childcraft books for arousing my interest.)

When I was older and learned that exponentiation actually existed, I was troubled by a curiosity of language. When people talk about 104, they say “ten to the power of four” — and yet 102, 103, 104 are collectively referred to as “powers of ten”. In one context, the number after the “of” is the exponent, and in the other context, it’s the base. Why is this? Isn’t mathematics supposed to be consistent?

If I could answer my childhood self, I’d point out that the same ambiguity crops up in everyday language. We say that x-ray vision is one of the powers of Superman, but we also say that Superman has the power of x-ray vision. When we use terms like “power of” we are translating mathematical notation into English, which has no use for consistency.

I not surprisingly came to speculate about an operation defined in terms of repeated exponentiation, another operation defined in terms of that, and so on ad infinitum. Some of you will find this familiar.

Let’s adopt the symbol ➀ for addition, ➁ for multiplication and ➂ for exponentiation, and let’s have ➊ for subtraction, ➋ for division and ➌ for the calculation of roots (although exponentiation actually has two inverse operations — the other being the calculation of logarithms — so we’ll leave that well alone).

Now we can specify operations using algebraic variables — and for that reason I referred to this system as operational algebra. I’ll use the symbol ✪ for “operation”, and write 2 [✪3] 5 to mean the same as 2 ➂ 5 which means the same as 25. For inverse operations I’ll insert a minus sign so that 2 [-✪2] 5 means the same as 2 ➋ 5 which means the same as 2/5. My original notation was handwritten, obviously, but this is a reasonable compromise. I won’t pretend it’s perfect.

For positive integers, higher-order operations can be defined as:

a [✪n+1] b = a [✪n] a [✪n] … [✪n] a {b terms}

But because exponentiation is non-associative, this presents us with a choice. If we bracket the expression from right to left — as in 2(2^(2^(2^…))) — then we are in the same realm as Knuth arrow notation (which I didn’t know about at the time). But if we bracket it from left to right — as in (((22)2)2)… — then because (ab)c=abc, it follows under this definition that ➃ can be extended to non-integer operands using the formula a ➃ b = a(a^(b-1)), and then a ➄ b can be defined for non-integer a. I always used the latter (left-associative) definition.

As for higher operations in this sequence, I was convinced at the time that they could all be extended to non-integer operands, but my reasoning was faulty.

The fun wasn’t all at the high end, though. Early on I recorded the trivial fact that there is no zeroth operation. That is, no operation can be consistently defined such that a + b = a ✪ a ✪ … ✪ a {b terms}.

I don’t propose to share everything I did with this system. For one thing, a lot of it was wrong, and for another, I threw my notes away years ago. But here is something worth finishing on. Recall that an inverse operation can be defined such that:

(a [-✪n] b) [✪n] b = a

Now observe:

2 ➊ 2 = 2 – 2 = 0

2 ➋ 2 = 2 / 2 = 1

2 ➌ 2 = sqrt(2) = 1.41421…

2 ➍ 2 = [solution of nn = 2] = 1.55961…

2 ➎ 2 = [solution of (n^(n^(n-1)) = 2] = 1.6498…

Based on this, it seems intuitively likely that if we could extend all higher operations to non-integer operands, then as n approaches infinity, we’d find that 2 [-✪n] 2 approaches 2, giving us a glimpse into the infinite operation. This intrigued me.

I abandoned my speculations in operational algebra around the time I left school and started university. I am vaguely aware that everything I explored (except the wrong bits) exists in established mathematics, if often subsumed under much more abstract generalisations. This makes no difference to the fact that a teenager had fun messing about.

As a final word: please don’t take from this blog post the idea that I’m good at maths. I lack the rigour to be good at it; I am merely playful.

There are several old posts on this blog about music that I composed when I was younger. But as a teenager in the nineties I also composed some fragments that never became complete songs, and I’ve not yet blogged about those.

So here is a selection. The lyrics are brimming with teenage angst, and are also rich in words I thought of as inherently poetic at the time. If any of them inspire you, you’re welcome to adapt them in your own creative works. All I ask is that you let me know.

Most of the recordings below were made expressly for this blog post, and are not polished performances. Their purpose is simply to demonstrate the tune.

The first one is the chronologically oldest, and here is a tune I recorded some years ago, followed by lyrics:

What I feel is a kind of tortureEvery moment reminds me of a thought that once occuredAnd I feel nothing less than tortureWhen I remember that the thought is too absurd.I cannot justify the course this tension’s leading;I wait unsatisfied, in pain and almost screaming.Waiting for some release from tortureTo release my mind, a thought that’s not absurdAnd I feel nothing less than tortureWhen I remember that the thought has not occured.

The next fragment is intended to sound like some old folk song. It’s rich in metaphor and open to interpretation. [Update: I have replaced the recording with sheet music. In time I may do the same with the others.]

There is no place so distant as my only world;There is no sound so faint as our most piercing scream.What shall I build, here where a thousand stones are hurled,And where the wind erodes away each worthless dream?

This one is about being abandoned by a valued friend. The second verse would have included the line, “And I’ll walk to infinity again today” (invoking a sense of aimlessness).

Regardless of the school you went to, if you did final year maths then you’ll relate to the following anecdote. Your stories might differ in specifics, but this is pretty much the archetype of what goes on in the classroom. (No doubt it also rings a bell if you’re a teacher.)

In a maths class, I asserted that the intelligence quotient of the classmate I was talking to was zero.

Said classmate replied that my IQ was zero POINT zero.

Me: “Yours is zero point zero times ten to the minus INFINITY.”

Him: “Yours is MINUS zero point zero times ten to the minus infinity.”

Me: “Yours is minus zero point zero times ten to the minus infinity *i*.”

Finally he concluded that my IQ was just i, a wholly imaginary quantity, and that at least his was real.

I tell this story as a gentle way to raise the topic of imaginary (and complex) numbers, which I’ve been musing on lately. I think my latest bout of musings were triggered by reading this, which I read via a chain of links that I can’t remember now, but would have included this.

The subject of my muse was why most attempts to explain the applications of complex numbers strike me as unilluminating.

Classroom banter doesn’t count as an application. Neither do pretty fractal patterns — they’re lovely, but like a game, the rules don’t have to mean anything. Neither is it remarkable that one abstraction can be used in the service of another abstraction, as in esoteric results from advanced mathematics. What counts are situations where it’s fruitful to take a quantity derived from real world measurements, and relate it to the square root of a negative number.

I never studied any applications of complex numbers myself, having not done any physics beyond first year university. But I’ve occasionally listened to, or read, people trying to explain how imaginary numbers are useful in their fields, and I almost invariably have no idea what they’re talking about. They might assert that imaginary numbers are useful for such-and-such, but in a way that leaves me no wiser about what said context has to do with squaring something and getting a negative result.

So it occurred to me that perhaps most explanations fail because they show the audience an imaginary-shaped peg without first cutting an imaginary-shaped hole. Let me illustrate the point with reference to negative numbers, and then relate it to imaginary/complex ones later on.

Once upon a time, negative numbers were widely regarded as an affront to common sense. Quantities less than nothing? Crazy. But applications soon became apparent, and those applications tended to fit very particular patterns. One can sensibly answer questions like: if you’re looking for a scenario that lends itself to being modelled with negative numbers, then what sort of characteristics should you look for?

In my metaphor, the answer to that question is the negative-shaped hole, and a particular scenario which has those characteristics is the negative-shaped peg. And here’s the answer: you look for a scenario where every quantity has a corresponding opposite. The opposite of having a particular speed is having the same speed in the opposite direction. The opposite of being so high off the ground is being so far under it (and, preferably, upside down). The opposite of having so much money is being so far in debt. And so on.

If you have a scenario in which various quantities have corresponding opposites, then you have a scenario that lends itself to being modelled with the aid of negative numbers. There’s your hole, and you’ll find any number of pegs to fit. Note also that negative numbers don’t measure anything mysterious; simply flip the coordinate system around and they become positive.

If we carry this pedagogical approach over to the imaginary case, the question becomes as follows. Suppose you’re looking for a scenario that lends itself to being modelled with imaginary numbers. What characteristics would you look for?

And for a pretty good answer, consider that in the world of imaginary numbers, there exists a function such that f(f(x)) = -x. In fact there are two such functions (multiplying by i and multiplying by -i), but the point is made by translating that formula into everyday language. It means there is something you can do to a quantity twice and always get back the opposite of what you started with.

That’s probably the simplest imaginary-shaped hole you can find. If you have a scenario in which there’s something you can do to a quantity twice and always get back the opposite of the quantity you started with, then you have a scenario that lends itself to being modelled with the aid of imaginary numbers. As for pegs, no problem. Try anything involving rotations and waves.

In the case of rotations, the thing you can do twice and end up with the opposite of what you started with is to rotate a quarter of the way around. In the case of waves, it’s to wait for a quarter of a wavelength to go by. Given such tools, if a physicist’s eyes roll at an elementary blunder in this blog post, you can calculate what that physicist will be looking at by the end of it. You can almost as simply calculate the same thing using old-fashioned real numbers, of course, but with a good i you can fit into a single formula what otherwise requires an algorithm. And you can show that the principle neatly carries over to finer intervals — less than a quarter of the whole.

I haven’t mentioned Euler’s equation — an oft-celebrated result of a very clever procedure that makes it meaningful to raise numbers to imaginary powers, and is typically taught in first year mathematics at university. The relevant consequence is that i can be re-written as e to the power of πi/2, and hence that i to the power of some number n is the same as e to the power of nπi/2 [note: π is pi — but my blog happens to use a font that doesn’t draw it very well]. Re-writing everything as exponents of e may make things more complicated for algebra, but simplifies things for calculus. But because I haven’t done any calculus since the 1990s (and wasn’t much good at it then), I won’t go there.

(Exponents of e are also neater if you measure angles in radians rather than quarter-revolutions, but I’m pretty sure the reason mathematicians measure angles in radians is because it leads to trigonometric functions having neat power series and hence Euler’s equation etc, so the causality is kind of inverted there.)

I’m definitely not going to mention quantum physics. If you’re interested, try a blog by someone who understands it. You didn’t come here for that.

I haven’t believed in God for a number of years now, but acknowledging that we are all shaped by our past, I occasionally like to talk about my former faith on this blog. Today I’d like to focus specifically on the subject of prayer.

I hope we can agree that the description of prayer found in The Last Hero by Terry Pratchett (“frightened people trying to make friends with the bully“) is a long way from the Christian understanding. But while it’s easy to talk about what prayer is not, discussing what it is requires grappling with paradoxes like why an omniscient God would need suggestions from humans. Back in the day I devoted considerable thought to theological questions like that, and I wrote some of those thoughts down in a document that I’m using as the main source for this post.

Prayer, according to what I consider the highest Christian understanding, is not about giving God suggestions. It’s more like tuning an aerial — that is, maintaining and refining the telepathic link between one’s self and God. Because God is understood in terms of moral perfection, and because there is no greater pursuit than to better ourselves morally, the Christian’s greatest aspiration is to think the thoughts of God. The point of prayer is to try to fill the mind with God’s thoughts, in part by putting into words the thoughts one believes to be “godlike” (such as compassionate wishes for other people). Deliberately focusing on godlike thoughts, it is thought, makes the mind more receptive to thoughts that come directly from God: the aerial-tuning analogy works pretty well here.

I grew up in rural South Australia, and when I was young my family often hosted Halloween parties. I particularly enjoyed carving the faces on the jack-o-lantern. Evil eyes, gorilla nostrils, and sharp teeth with long canines were my style. Incidentally, our jack-o-lanterns were watermelons, because they’re bigger than pumpkins.

Another favourite activity was the doughnut game. It seems not many people know about this, and I don’t know where we picked it up from. You’ll find some relevant hits if you search the Internet for “halloween doughnut game”, but they’re all tragically misinformed because they leave out the most important part (the chocolate sauce).

In a comment on Stan Carey’s blog, I mentioned a regular spelling system that I once invented for my dialect of English.

This was never intended as a cure for the irregularities of English spelling. I’ve always been happier inventing things that are pointless but fun over anything of practical use, so I made it in many ways the opposite of what you’d want from a workable spelling reform. It’s very specific to my own dialect, contains complicated and sometimes ambiguous rules, and has the look of a completely alien language. These qualities give it a certain aesthetic appeal, but certainly not a practical one.

On Tuesday I attended the funeral of Pam Marlow, a very good friend who recently died from ovarian cancer, aged sixty.

Pam’s family and mine have always been close, my parents having known Pam and her husband Phil since long before I was born. My earliest memories of the Marlows come from the mid eighties, when our families would camp together in the sandhills every Easter.

Pam has done a lot for me in recent years. I don’t drive, and Pam has taken me grocery shopping every fortnight or so. These shopping trips were more than just a personal favour; they were also social meetups, and Pam clearly enjoyed the conversations that we had on the way and back.

Pam lead a busy life, so it was not often that we stopped for a coffee after shopping, but we did so occasionally. One way in which I sometimes tried to repay her was with tickets to Fringe Festival performances and such things. For example, she accompanied me to Mediaeval Magic by the Luna Vocal Ensemble in 2008, and to Dave Bloustien’s Complete History of Western Philosophy in 2010.